This section proposes a discrete, direct methodology for the coupling of 3D lattice Boltzmann equations with dynamic (ODE-based) 0D models which, as described before, are often found in inflow and outflow conditions for cardiovascular configurations. The hybrid ODE-Dirichlet system governed by Equation (REF ) represents a most complex form of the myriad such formulations found in cardiovascular modeling, and its particular treatment is discussed in Section REF  (although the method proposed in what follows is straightforwardly applicable to other ODE-based 0D boundary equations such as Windkessel models). The general strategy for such multidimensional coupling of the solver presented in this work is based on the non-equilibrium extrapolation method [1]}, [2]}, which has been introduced as an alternative to typical “bounce-back” methods [3]} used to implement (given) pressure and velocity boundary conditions for LB methods in order to preserve a consistent order-of-accuracy between the boundaries and the order-of-accuracy inherent to LB formulations [2]}. In particular, such a method is ideally suited for curved fluid boundaries [5]}, [6]} (such as those provided by fluid-structure interfaces of interest in this work), where the physical boundary need not coincide with the regular fluid lattice.
